This invention relates generally to computer systems, and more particularly, to a computational device that evaluates a multiple integral having two or more dimensions.
Multiple integrals are well known as having utility in a large variety of scientific and analytical situations, particularly in determining cumulative effects. The concept that forms the basis of multiple integration is directed toward producing a value of a multidimensional cumulative effect, including total content, total value, total benefit, etc. In scientific and analytical contexts, such totals are obtained with precision when a precise formula is available expressing the quantity that is present or can be expected in each small portion of a region under consideration.
In the specific illustration of applying multiple integration to the problem of determining total coal resources in a given area, illustratively the United States, a specific formula may be known in the form of a density, illustratively expressing the quantity of coal per square mile, or other small portion of the region being considered. In the case of coal, one can also consider the amount of coal at various depths below the surface and obtain a density in tonnage per cubic mile. The formula, or function, provides one number that corresponds to the coal density at each location. Thus, to describe such a location, one needs two numbers, illustratively latitude and longitude. It can therefore be said that the coal density is a function of two variables. However, if a third number is required, such as depth, the function would be of the type having three variables.
In practice, one often encounters functions or three, four, or more variables that are to be integrated. For example, the determination of the total production of a factory over a year is a complicated integration process. In such a process, the density corresponds to a rate of production under various conditions. Description of such a process may require a formula having up to seven variables, which may correspond to time, location in the factory, variable supplies of raw materials, variable power supply, etc. In this manner, one can be required to integrate a function of any number of variables.
If a formula is known precisely, one can occasionally apply mathematics in the discipline of integral calculus to determine the total, or integral. However, such a situation occurs only relatively rarely and one usually must apply "brute force" requiring either hand calculators or computers. A brute force approach to a specific example is illustrated by assuming the simple formula: EQU d=7X+2Y (tons per square mile),
which represents the density of coal across a square field, one mile on each side, in terms of distances X and y from two adjacent sides. Here X and Y can be measured in miles. Thus, at the illustrative point where X=0.62 and Y=0.73, EQU d=7(0.62)+2(0.73)=5.80.
The foregoing evaluation of the formula indicates that, for a small piece of field around the point where X=0.62 and Y=0.73, one could expect to find 5.80.times.A tons, where A is the area of the piece. One then adds all these small contributions to find the total coal deposit beneath the field.
The foregoing method seems crude, but is the basis for all integration. Increasing accuracy is achieved by increasing the number of small pieces and decreasing the size of each piece. It is therefore evident that application of the brute force method to functions of three, four, or more variables results in a very rapid increase in the number of small pieces required for high accuracy.
It has long been known that accurate results can be obtained more quickly by using averages, or weighted averages, of function values at selected points. For example, assuming that a small square area is to be analyzed, d.sub.0 is the density at the center of the square, d.sub.1, . . . ,d.sub.4 are the densities at the four corners, and d.sub.5, . . . ,d.sub.8 are the densities at the midpoints of the sides. Using Simpson's Rule, the weighted average in the form: ##EQU1## results in a relatively accurate estimate of the average density over the square. One can obtain the total coal tonnage by multiplying the area of the field, which is one square mile in the specific example, by the average density obtained using Simpson's Rule.
It is, therefore, an object of this invention to provide a device for evaluating multiple integrals.
It is a further object of this invention to incorporate selected averaging rules in a device for evaluating multiple integrals.
It is another object of this invention to provide a device for evaluating multiple integrals which achieves a high degree of accuracy at relatively low cost.
It is also an object of this invention to provide a device which can evaluate multiple integrals of functions having up to 8 variables.
It is another object of this invention to provide a device for evaluating multiple integrals wherein a user can select a desired level of accuracy.
It is a still further object of this invention to provide a device to evaluate multiple integrals wherein a user can limit computing time in response to a cost-benefit analysis.
It is yet an additional object of this invention to provide a plurality of methods of operating a device for evaluating multiple integrals Wherein a plurality of solution sets are provided, each subsequent method producing a lower absolute error value when applied to a test function.